June 14, 2013Location: Janskerkhof 15a (Utrecht), room 202
We investigate global logarithmic asymptotics of ground states for a family of quantum mean field models with a transverse component. Stochastic representation enables a reinterpretation of ground states in terms of quasi-stationary measures for Markov chains in killing potentials. The subsequent analysis is built upon a combination of large deviation and weak KAM techniques. In a fairly general situation limiting ground states are fixed points of associated Lax–Oleinik semigroups and, accordingly, viscosity solutions of the corresponding Hamilton–Jacobi equations. Such a description does not yield unicity at critical strengths of transverse fields, and first order transitions can occur. The spin 1/2 case with general (e.g. p-body) mean field interactions is worked out in detail.
Based on recent joint work with Anna Levit.
In this talk, we will discuss the continuity/discontinuity of the phase transition of the planar Potts models. More precisely, we will show that the transition is continuous, meaning that there is no spontaneous magnetization at criticality, for ferromagnetic Potts models on the square lattice with 2, 3 and 4 colors. The proof is based on two important ingredients. The first one is the use of parafermionic observables and discrete holomorphicity to show that the correlation length diverges at criticality. The second one is a new theorem relating different notions of continuous phase transition (absence of spontaneous magnetization, divergence of the correlation length, divergence of the susceptibility, uniqueness of Gibbs states, RSW type results, etc). This second ingredient enables us to obtain quantitative information on the critical phase. We will also explain what is missing in order to prove Baxter's conjecture that the phase transition is discontinuous if and only if there are strictly more than 4 colors. Joint work with V. Tassion and V. Sidoravicius.